1. Bring stock to a boil in a medium saucepan over high heat. Reduce heat to low to keep stock hot.

2. Melt 4 tbsp. of the butter in a medium heavy-bottomed pot over medium-high heat. Add onions and cook, stirring with a wooden spoon, until they begin to brown, about 5 minutes. Add uncooked pasta and carrots and cook, stirring often, until pasta is lightly toasted, about 5 minutes.

3. Add 1 cup of the hot stock at a time to the pasta, stirring constantly; wait until almost all the stock has been absorbed before adding more. Continue cooking and adding stock (you may have some stock left over) until pasta is tender but firm to the bite, 10–15 minutes.

4.Remove pot from heat and stir in the remaining 1 tbsp. butter and the parmigiano. Season to taste with salt and pepper. Transfer to a warm serving dish and garnish with parsley. Serve with more freshly grated parmigiano, if you like.